We now study \(B\) -field driven ZBPs in a different parameter regime: we again consider the evolution of in-gap ABSs far from the gap edge at \(B=0\)  but close to a singlet-doublet transition mediated by \(V_{\textrm {bg}}\) , see the three colored diamonds in Fig.  REFA. Our results are presented in Fig.  REFE. The three subpanels show the evolution in magnetic field of the same pair of ABSs but at slightly different backgate values (see also Fig. S7 in the SI).
The closer the backgate is to the singlet-doublet zero-energy crossing, the smaller the ABS energy is. For increasing magnetic fields, this \(B=0\)  energy influences the particular \(B\)  value at which the excitation reaches zero energy (compare the panels in Fig.  REFE as one moves from left to right).
This gate dependence allows to tune the value of \(B\)  at which the ZBP emerges. For \(V_{\textrm {bg}}=-2.705 {V}\) , in particular (blue diamond),
the ZBP appears at the beginning of the first lobe and persists throughout its full extent. The lower panels in Fig.  REFE show numerics in this regime, whose overall agreement is excellent. This showcases how ABSs may produce a strong ZBP persisting through a large range of magnetic fields [1]}.
After the ZBP, the GS changes from singlet to doublet. This spin-polarised YSR doublet follows again Eq. (REF ) (see SI).
